Schreiber Synthetic quantum theory in higher cohesive toposes

This summer I will be giving a talk at the 2013 CMS summer meeting (Halifax, June 2013) in the session Progress in higher categories.

• Urs Schreiber,

  Synthetic quantum theory in higher cohesive toposes

on Higher geometric prequantum theory in the context of differential cohomology in a cohesive topos.

Abstract

There are two traditional mathematical formalizations of quantum physics via quantization: “algebraic deformation quantization” and “geometric quantization”. I discuss how the latter has a natural axiomatic (“synthetic”) formulation in those higher toposes which are equipped with two adjoint triples of idempotent higher modalities – a joint refinement of what Lawvere had called “synthetic differential geometry” and “cohesion”. This yields synthetic re-derivations of classical and of more recent results in geometric quantization, and lifts them from quantum mechanics to local (“extended”) quantum field theory in higher dimension. I close by indicating examples of Chern-Simons type field theories. The example in dimension 2 recovers C-star algebraic deformation quantization as its “holographic dual”; the example in dimension 3 holographically recovers twisted equivariant K-theory. The example in dimension 7 relates to current questions in non-perturbative string theory.

For more material see at Higher geometric prequantum theory.